Is a Third-Rank Tensor Viable for Expressing Manifold Curvature?

[copernicus1]

Hi, I am curious about expressing the curvature of a manifold using a third-rank tensor. The fourth order Riemann tensor can be contracted to give the second order Ricci tensor and zeroth order Ricci scalar, but is there no way of obtaining a third- or first-order tensors, or would this simply not make sense for some reason?

Thanks!

 

[jvicens]

Hi there,

Thank you for your question. Expressing the curvature of a manifold using a third-rank tensor is possible and has been done in certain cases. The third-rank tensor is known as the Cotton tensor and is defined as a contraction of the Riemann tensor with the dual of the Ricci tensor. It can also be written in terms of the Weyl tensor, which is a second-rank tensor, and the Ricci tensor.

The reason why third- or first-order tensors are not commonly used for expressing the curvature of a manifold is that they do not provide any additional information that cannot already be obtained from the Riemann or Ricci tensors. In fact, the Riemann tensor contains all the necessary information about the curvature of a manifold. The Ricci tensor and scalar are derived from the Riemann tensor and provide a more compact representation of the curvature.

However, in certain cases, the Cotton tensor may be useful for studying specific properties of a manifold, such as conformal invariance. So while it is possible to obtain third- or first-order tensors, they may not necessarily be needed in most cases.

I hope this helps clarify your question. Let me know if you have any further inquiries.